[Solved] Remarks in equations with arrows

Mathy Asks: Remarks in equations with arrows
In huge calculations it’s often very helpful as reader to have nice remarks about which manipulation was made. Also, other remarks in equations can help alot, e.g., this one


enter image description here

which I like more than the ^-convention. I am generating the above with the following code

Code:
vertarrowbox{e_j}{$i$textsuperscript{th} spot}

where

Code:
newcommandvertarrowbox[3][2ex]{%
    begin{array}[t]{@{}c@{}} #2 \[-0.7ex]
        leftuparrowvcenter{hrule height #1}right.kern-nulldelimiterspace\[-1.2ex]
        makebox[0pt]{scriptsize#3}
    end{array}%
}

My goal now is to achieve something similar. I am looking for something like this


enter image description here

to denote that the exchange of the two variables gives a minus sign.

I wonder how I could implement this. I am not sure if tikz-cd is the way to go (I don’t have experience with tikz-cd in equations like this and don’t know if it is even possible to use it this way).

Thank you for your ideas!

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